RENDICONTI DEL SEMINARIO MATEMATICO

Zero shear viscosity limit in compressible isentropic fluids 43So, recalling the pointwise boundedness of ρ, one immediately gets from (29) and (30)that u 2 ǫ ⇀ u2 weakly in L 2 (Q T ), which combined with (24) shows thatu ǫ → u strongly in L 2 (Q T ).Therefore, by interpolation and (20), we infer that(31) u ǫ → u strongly in L s (Q T ), ∀ s < 6.Now, we use and adapt the techniques in [39, 14, 16] (also cf. [29]) to prove thefollowing strong convergence of ρ ǫ .LEMMA 5.0∫ Tρ ǫ → ρ strongly in L 1 (Q T ), as ǫ → 0.Proof. First, multiplying (2) by φ ∈ C0 ∞() and integrating over (r 1, x), then multiplyingthe resulting equation by ψ(t)ρ ǫ , ψ(t) ∈ C0 ∞ (0, T), and integrating over(0, T) × , we obtain after a straightforward calculation that∫ T ∫ [ (ψ(t) ρ ǫ aρǫ γ − (λ + 2ǫ) ∂ x u ǫ + u )]ǫφdxdt0 x∫ T ∫ ∫ x∫ T ∫ ∫= ψ ′ 1 x(t) ρ ǫ ρ ǫ u ǫ φdξdxdt − ψ0 r 10 x ρ ǫu ǫ ρ ǫ u ǫ φdξdxdtr 1∫ T ∫ ∫ x∫ T ∫ ∫ x+ ψ ρ ǫ (aρǫ γ + ρ ǫu 2 ǫ )φ ρ ǫ u 2 ǫξ dξdxdt − ψ ρ ǫ φdξdxdt0 r 1 0 r 1ξ∫ T ∫ ∫ xρ ǫ vǫ2 + ψ ρ ǫ φdξdxdtξ(32)−(λ + 2ǫ)0r 1∫ψ∫ x(ρ ǫ ∂ ξ u ǫ + u ǫr 1ξ)φ ξ dξdxdt,where H := aρǫ γ − (λ + 2ǫ)(∂ x u ǫ + u ǫ /x) is so-called the effective viscous pressurewhich possesses some smoothing property and plays an important role in the existenceproof of global weak solutions to the multidimensional compressible Navier-Stokesequations, cf. [39, 29, 14, 15].Now, passing to the limit in (32) as ǫ → 0, and making use of (22)–(29) and(31), we see that=(33)∫ T0∫ T0++∫ψ(t)∫ψ ′ (t)∫ T0∫ T0∫ψ∫ψ[ (aρ γ+1 − λ∫ xρρρr 1∫ x∫ x)]φdxdtρu x + ρux∫ T ∫ ∫1 xρuφdξdxdt − ψ0 x ρu ρuφdξdxdtr 1∫ T ∫ ∫ x(aρ γ + ρu 2 ρu 2)φ ξ dξdxdt − ψ ρr 1 0 r 1ξ φdξdxdtρv 2 ∫ T ∫ ∫ xr 1ξ φdξdxdt − λ ψ ρ(u ξ + u )φ ξ dξdxdt.0 r 1ξ